Problem: Emily is 4 times as old as Jessica. Six years ago, Emily was 7 times as old as Jessica. How old is Jessica now?
Solution: We can use the given information to write down two equations that describe the ages of Emily and Jessica. Let Emily's current age be $e$ and Jessica's current age be $j$ The information in the first sentence can be expressed in the following equation: $e = 4j$ Six years ago, Emily was $e - 6$ years old, and Jessica was $j - 6$ years old. The information in the second sentence can be expressed in the following equation: $e - 6 = 7(j - 6)$ Now we have two independent equations, and we can solve for our two unknowns. Because we are looking for $j$ , it might be easiest to use our first equation for $e$ and substitute it into our second equation. Our first equation is: $e = 4j$ . Substituting this into our second equation, we get: $4j$ $-$ $6 = 7(j - 6)$ which combines the information about $j$ from both of our original equations. Simplifying the right side of this equation, we get: $4 j - 6 = 7 j - 42$ Solving for $j$ , we get: $3 j = 36.$ $j = 12$.